Problem 71 Simplify the given expression. ... [FREE SOLUTION]

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Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 5: Problem 71

Simplify the given expression. $$\left(2 x^{2}\right)(5 x y)\left(-y^{2}\right)+x^{3} y^{3}$$

Short Answer

Expert verified

-9 x^{3} y^{3}

Step by step solution

- Rewrite the Expression Without Parentheses

Analyze the given expression: \there \texpected \tis \t\[\t(2 x^{2})(5 x y)(-y^{2}) + x^{3} y^{3}\t\].

- Simplify Each Part Separately

First part \t: \t(2 x^{2})(5 x y)(-y^{2}):\tMultiply numerical coefficients first: \t2 * 5 * (-1) = -10.\t\tCombine powers of \tx \tand \ty: \tx^{2} * x = x^{3} \tand \ty * y^{2} = y^{3}.\tThe simplified expression for the first part becomes: \t\[\t-10 x^{3} y^{3}.\t\]

- Combine Like Terms

Add the simplified first part and the second part of the original expression: -10 x^{3} y^{3} + x^{3} y^{3}.\tCombine the like terms \[-10 x^{3} y^{3}+x^{3} y^{3}= $-10+1$ x^{3} y^{3} = -9 x^{3} y^{3}.\t\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplification

Simplification involves making algebraic expressions easier to work with by combining like terms and applying arithmetic operations. It usually makes equations and expressions more manageable. In the given exercise, we first break down the complicated expression by removing parentheses and then simplifying each part individually. This method helps transform a complex expression into its simplest form, making it easier to understand and solve.

Exponents

Exponents represent repeated multiplication of a number by itself. In the given exercise, we have terms like $x^{2}$ and $y^{2}$. Exponents obey certain rules, such as $a^{m} \times a^{n} = a^{m+n}$. For instance, $x^{2} \times x$ becomes $x^{3}$. Similarly, $y \times y^{2} = y^{3}$. By using these rules, the given expression is simplified accurately.

Combining Like Terms

Combining like terms combines terms that have the same variables raised to the same powers. In our example, we combine $-10 x^{3} y^{3}$ and $x^{3} y^{3}$. Like terms here have the same variables $(x^{3} y^{3})$. So, $-10 + 1$ gives $-9 x^{3} y^{3}$. Combining like terms simplifies the expression significantly by reducing the number of terms.

Multiplication of Variables

Multiplication of variables requires multiplying their coefficients and then using the properties of exponents. For example, in $(2 x^{2})(5 x y)(-y^{2})$, we first multiply the coefficients: \2 \times 5 \times -1 = -10\. Next, we multiply the variables: \x^{2} \times x = x^{3}, y \times y^{2} = y^{3}\. This gives $-10 x^{3} y^{3}$. Such steps are crucial for simplifying expressions.

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91影视

Simplify the given expression. $$\left(2 x^{2}\right)(5 x y)\left(-y^{2}\right)+x^{3} y^{3}$$

Short Answer

Step by step solution

- Rewrite the Expression Without Parentheses

- Simplify Each Part Separately

- Combine Like Terms

Key Concepts

Simplification

Exponents

Combining Like Terms

Multiplication of Variables

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